Download Chapter 8 Relativistic Electromagnetism

Chapter 8
Relativistic Electromagnetism
In which it is shown that electricity and magnetism can no more be separated
than space and time.
8.1
Magnetism from Electricity
Our starting point is the electric and magnetic fields of an infinite straight
wire, which are derived in most introductory textbooks on electrodynamics,
such as Griffiths [3], and which we state here without proof.
The electric field of an infinite straight wire with charge density λ points
away from the wire with magnitude
E=
λ
2π0 r
(8.1)
where r is the perpendicular distance from the wire and 0 is the permittivity constant. The magnetic field of such a wire with current density I has
magnitude
µ0 I
B=
(8.2)
2πr
with r as above, and where µ0 is the permeability constant, which is related
to 0 by
1
0 µ 0 = 2
(8.3)
c
(The direction of the magnetic field is obtained as the cross product of the
direction of the current and the position vector from the wire to the point in
question.)
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CHAPTER 8. RELATIVISTIC ELECTROMAGNETISM
~ on a
We will also need the Lorentz force law, which says that the force F
test particle of charge q and velocity ~v is given by
~ = q(E
~ +~v × B)
~
F
(8.4)
~ and B
~ denote the electric and magnetic fields (with magnitudes E
where E
and B, respectively).
Consider an infinite line charge, consisting of identical particles of charge
ρ, separated by a distance `. This gives an infinite wire with (average) charge
density
ρ
(8.5)
λ0 =
`
Suppose now that the charges are moving to the right with speed
u = c tanh α
(8.6)
Due to length contraction, the charge density seen by an observer at rest
increases to
ρ
λ=
= λ0 cosh α
(8.7)
`
cosh α
Suppose now that there are positively charged particles moving to the
right, and equally but negatively charged particles moving to the left, each
with speed u. Consider further a test particle of charge q situated a distance
r from the wire and moving with speed
v = c tanh β
(8.8)
to the right. Then the net charge density in the laboratory frame is 0, so
that there is no electrical force on the test particle in this frame. There is of
course a net current density, however, namely
I = λu + (−λ)(−u) = 2λu
(8.9)
What does the test particle see? Switch to the rest frame of the test
particle; this makes the negative charges appear to move faster, with speed
u− > u, and the positive charges move slower, with speed u+ < u. The
relative speeds satisfy
u+
= tanh(α − β)
c
u−
= tanh(α + β)
c
(8.10)
(8.11)
8.1. MAGNETISM FROM ELECTRICITY
45
resulting in current densities
λ± = λ cosh(α ∓ β) = λ(cosh α cosh β ∓ sinh α sinh β)
(8.12)
resulting in a total charge density of
λ0 = λ + − λ −
= −2λ0 sinh α sinh β
= −2λ tanh α sinh β
(8.13)
(8.14)
(8.15)
According to (8.1), this results in an electric field of magnitude
E0 =
λ0
2π
(8.16)
which in turn leads to an electric force of magnitude
F 0 = qE 0 = −
λ
q tanh α sinh β
π0 r
λu
qv cosh β
π0 c2 r
µ0 I
= −
qv cosh β
2πr
= −
(8.17)
(8.18)
(8.19)
To relate this to the force observed in the laboratory frame, we must
consider how force transforms under a Lorentz transformation. We have 1
0
~ 0 = d~p
F
dt0
(8.20)
and of course also
~ = d~p
(8.21)
F
dt
But since in this case the force is perpendicular to the direction of motion,
we have
d~p = d~p0
(8.22)
and since dx0 = 0 in the comoving frame we also have
dt = dt0 cosh β
1
(8.23)
This is the traditional notion of force, which does not transform simply between frames.
As discussed briefly below, a possibly more useful notion of force is obtained by differentiating with respect to proper time.
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CHAPTER 8. RELATIVISTIC ELECTROMAGNETISM
Thus, in this case, the magnitudes are related by
F =
µ0 I
F0
=−
qv
cosh β
2πr
(8.24)
But this is just the Lorentz force law
~ = q~v × B
~
F
(8.25)
~ given by (8.2)!
with B = |B|
We conclude that in the laboratory frame there is a magnetic force on the
test particle, which is just the electric force observed in the comoving frame!
8.2
Lorentz Transformations
We now investigate more general transformations of electric and magnetic
fields between different inertial frames. Our starting point is the electromagnetic field of an infinite flat metal sheet, which is derived in most introductory
textbooks on electrodynamics, such as Griffiths [3], and which we state here
without proof.
The electric field of an infinite metal sheet with charge density σ points
away from the sheet and has the constant magnitude
σ
|E| =
(8.26)
20
The magnetic field of such a sheet with current density ~κ has constant magnitude
µ
|B| = |~κ|
(8.27)
2
and direction determined by the right-hand-rule.
Consider a capacitor consisting of 2 horizontal (z = constant) parallel
plates, with equal and opposite charge densities. For definiteness, take the
charge density on the bottom plate to be σ0 , and suppose that the charges
are at rest, that is, that the current density of each plate is zero. Then the
electric field is given by
~ 0 = E0~ = σ0 ~
E
(8.28)
0
between the plates and vanishes elsewhere. Now let the capacitor move to
the left with velocity
~u = −u~ı = −c tanh α ~ı
(8.29)
8.2. LORENTZ TRANSFORMATIONS
47
Then the width of the plate is unchanged, but, just as for the line charge
(8.7), the length is Lorentz contracted, which decreases the area, and hence
increases the charge density. The charge density (on the bottom plate) is
therefore
σ = σ0 cosh α
(8.30)
But there is now also a current density, which is given by
~κ = σ~u
(8.31)
on the lower plate. The top plate has charge density −σ, so its current
density is −~κ. Then both the electric and magnetic fields vanish outside the
plates, whereas inside the plates one has
~ = E y~ = σ ~
(8.32)
E
0
~ = B z ~k = −µ0 σu ~k
(8.33)
B
which can be rewritten using (8.29) and (8.30) in the form
E y = E0 cosh α
B z = cB0 sinh α
(8.34)
(8.35)
For later convenience, we have introduced in the last equation the quantity
1
E0
(8.36)
c2
which does not correspond to the magnetic field when the plate is at rest —
which of course vanishes since ~u = 0.
The above discussion gives the electric and magnetic fields seen by an
observer at rest. What is seen by an observer moving to the right with
speed v = c tanh β? To compute this, first use the velocity addition law to
compute the correct rapidity to insert in (8.35), which is simply the sum of
the rapidities α and β!
~ 0 and a magnetic
The moving observer therefore sees an electric field E
~ 0 . From (8.35), (8.36), and the hyperbolic trig formulas (4.5) and (4.6),
field B
we have
B0 = −µ0 σ0 = −µ0 0 E0 = −
E 0y =
=
=
=
E0 cosh(α + β)
E0 cosh α cosh β + E0 sinh α sinh β
E0 cosh α cosh β − c2 B0 sinh α sinh β
E y cosh β − cB z sinh β
(8.37)
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CHAPTER 8. RELATIVISTIC ELECTROMAGNETISM
and similarly
B 0z = cB0 sinh(α + β)
= cB0 sinh α cosh β + cB0 cosh α sinh β
1
= B z cosh β − E y sinh β
c
(8.38)
Repeating the argument with the y and z axes interchanged (and being
careful about the orientation), we obtain the analogous formulas
E 0z = E z cosh β + cB y sinh β
1
B 0x = B y cosh β + E z sinh β
c
(8.39)
(8.40)
Finally, by considering motion perpendicular to the plates one can show [3]
E 0x = E x
(8.41)
and by considering a solenoid one obtains [3]
B 0x = B x
(8.42)
Equations (8.37)–(8.42) describe the behavior of the electric and magnetic
fields under Lorentz transformations. These equations can be nicely rewritten
in vector language by introducing the projections parallel and perpendicular
to the direction of motion of the observer, namely
~
~ k = ~v · E ~v
E
~v ·~v
~
~ k = ~v · B ~v
B
~v ·~v
(8.43)
(8.44)
and
~⊥ = E
~ −E
~k
E
~⊥ = B
~ −B
~k
B
(8.45)
~k
~0 = E
E
k
~0 = B
~k
B
k
(8.47)
(8.46)
We then have
(8.48)
49
8.3. TENSORS
and
~ ⊥ cosh β
~ ⊥ − 1 ~v × E
B
2
c
~ ⊥ +~v × B
~ ⊥ cosh β
= E
~0 =
B
⊥
~0
E
⊥
8.3
Tensors
8.3.1
Vectors
(8.49)
(8.50)
In the previous chapter, we used 2-component vectors to describe spacetime,
with one component for time and the other for space. In the case of 3 spatial
dimensions, we use 4-component vectors, namely
x0
ct
 x1 
x

 
xν = 
 2 =  
x 
y 
3
x
z




(8.51)
These are called contravariant vectors, and their indices are written “upstairs”, that is, as superscripts.
Just as before, Lorentz transformations are hyperbolic rotations, which
must now be written as 4 × 4 matrices. For instance, a “boost” in the x
direction now takes the form
cosh β
ct0
 − sinh β
 x0 



 0 =

y 
0
z0
0



− sinh β
cosh β
0
0
0
0
1
0
0
ct
x
0
 
 
0 y 
1
z


(8.52)
A general Lorentz transformation can be written in the form
x0µ = Λµ ν xν
(8.53)
where Λµ ν are (the components of) the appropriate 4 × 4 matrix, and where
we have adopted the Einstein summation convention that repeated indices,
in this case ν, are to be summed from 0 to 3. In matrix notation, this can
be written as
x0 = Λx
(8.54)
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CHAPTER 8. RELATIVISTIC ELECTROMAGNETISM
Why are some indices up and others down? In relativity, both special and
general, it is essential to distinguish between 2 types of vectors. In addition
to contravariant vectors, there are also covariant vectors, often referred to
as dual vectors. The dual vector associated with xµ is 2




−ct
−x0
 x 
 x 





(8.55)
xµ =  1  = 
 y 
 x2 
z
x3
We won’t have much need for covariant vectors, but note that the invariance
of the interval can be nicely written as
xµ xµ = −c2 t2 + x2 + y 2 + z 2
= x0µ x0µ
(8.56)
(Don’t forget the summation convention!) In fact, this equation can be taken
as the definition of Lorentz transformations, and it is straightforward to
determine which matrices Λµ ν are allowed.
Taking the derivative with respect to proper time leads to the 4-velocity
dxµ
dxµ dt
=
dτ
dt dτ
It is often useful to divide these into space and time in the form
uµ =
cγ
u=
~v γ
c cosh β
=
v̂ c sinh β
(8.57)
(8.58)
where v̂ is the unit vector in the direction of ~v . Note that the 4-velocity is a
unit vector in the sense that
1
uµ uµ = −1
(8.59)
c2
The 4-momentum is is simply the 4-velocity times the rest mess, that is
µ
µ
p = mu =
1
c
E
~p
=
mcγ
m~v γ
mc cosh β
=
v̂ mc sinh β
(8.60)
and note that
pµ pµ = −m2 c2
(8.61)
which is equivalent to our earlier result
E 2 − p 2 c2 = m 2 c4
2
Some authors use different conventions.
(8.62)
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8.4. THE ELECTROMAGNETIC FIELD
8.3.2
Tensors
Roughly speaking, tensors are like vectors, but with more components, and
hence more indices. We will only consider one particular case, namely rank
2 contravariant tensors, which have 2 “upstairs” indices. Such a tensor has
components in a particular reference frame which make up a 4 × 4 matrix,
namely

 00
T
T 01 T 02 T 03
 T 10 T 11 T 12 T 13 


(8.63)
 20
T µν = 
T
T 21 T 22 T 23 
T 30 T 31 T 32 T 33
How does T transform under Lorentz transformations? Well, it has two
indices, each of which must be transformed. This leads to a transformation
of the form
T 0µν = Λµ ρ Λν σ T ρσ = Λµ ρ T ρσ Λν σ
(8.64)
where the second form (and the summation convention!) leads naturally to
the matrix equation
T 0 = ΛT Λt
(8.65)
where t denotes matrix transpose.
Further simplification occurs in the special case where T is antisymmetric,
that is
T νµ = −T µν
(8.66)
so that the components of T take the form
T µν
8.4
0
a
 −a
0
=

 −b −f
−c e

b
c
f −e 


0
d 
−d 0

(8.67)
The Electromagnetic Field
Why have we done all this? Well, first of all, note that, due to antisymmetry,
T has precisely 6 independent components. Next, compute T 0 , using matrix
multiplication and the fundamental hyperbolic trig identity (4.4). As you
52
CHAPTER 8. RELATIVISTIC ELECTROMAGNETISM
should check for yourself, the result is
T 0µν
0
 −a0

= 0
 −b
−c0

a0
0
−f 0
e0
b0
f0
0
−d0
c0
−e0 


d0 
0

(8.68)
where
a0
b0
c0
d0
e0
f0
=
=
=
=
=
=
(8.69)
(8.70)
(8.71)
(8.72)
(8.73)
(8.74)
a
b cosh β − f sinh β
c cosh β + e sinh β
d
e cosh β + c sinh β
f cosh β − b sinh β
The first 3 of these are the transformation rule for the electric field, and the
remaining 3 are the transformation rule for the magnetic field!
We are thus led to introduce the electromagnetic field tensor, namely
F uv
8.4.1
0
 1 x
 −c E
=
 1 y
 −c E
− 1c E z

1
c
Ex
0
−B z
By
1
c
Ey
Bz
0
−B x
1
c
Ez

−B y 

Bx 

0

(8.75)
Maxwell’s equations
Maxwell’s equations in vacuum (and in MKS units) are
1
ρ
0
~ ·B
~ = 0
∇
~
~ ×E
~ = − ∂B
∇
∂t
(8.77)
~
~ ×B
~ = µ0 J
~ + µ0 0 ∂E
∇
∂t
(8.79)
~ ·E
~ =
∇
(8.76)
(8.78)
8.4. THE ELECTROMAGNETIC FIELD
53
~ is the current density, and the constants
where ρ is the charge density, J
µ0 and 0 satisfy (8.3). Equation (8.76) is just Gauss’ Law, (8.78) is Faraday’s equation, and (8.79) is Ampère’s Law corrected for the case of a timedependent electric field. We also have the charge conservation equation
~ ·J
~ = − ∂ρ
∇
∂t
(8.80)
~ = q(E
~ +~v × B)
~
F
(8.81)
and the Lorentz force law
The middle two of Maxwell equations are automatically solved by intro~ and defining
ducing the scalar potential Φ and the vector potential A
~ = ∇
~ ×A
~
B
~
~ = − ∂A − ∇Φ
~
E
∂t
8.4.2
(8.82)
(8.83)
Tensor Version of Maxwell’s Equations
Consider the following derivatives of F :
∂F µ0 ∂F µ1 ∂F µ2 ∂F µ3
∂F µν
=
+
+
+
∂xν
∂t
∂x
∂y
∂z
(8.84)
This corresponds to four different expressions, one for each value of µ. For
µ = 0, we get
0+
1~ ~
ρ
1 ∂E x 1 ∂E y 1 ∂E z
+
+
= ∇
= cµ0 ρ
·E =
c ∂x
c ∂y
c ∂z
c
c0
(8.85)
where Gauss’ Law was used to get the final two equalities. Similarly, for
µ = 1 we have
1 ∂E x
∂B z ∂B y
− 2
+0+
−
(8.86)
c ∂t
∂y
∂z
and combining this with the expressions for µ = 2 and µ = 3 yields the
left-hand-side of
~
1 ∂E
~ = µ0J
~
− 2
+ ∇ ×B
(8.87)
c ∂t
54
CHAPTER 8. RELATIVISTIC ELECTROMAGNETISM
where the right-hand-side follows from Ampère’s Law. Combining these
equations, and introducing the 4-current density
µ
J =
cρ
~
J
(8.88)
leads to
∂F µν
~µ
= µ0J
(8.89)
ν
∂x
which is equivalent to the two Maxwell equations with a physical source,
namely Gauss’ Law and Ampère’s Law.
Furthermore, taking the (4-dimensional!) divergence of the 4-current density leads to
∂J µ
∂ ∂F µν
µ0 µ =
=0
(8.90)
∂x
∂xµ ∂xν
since there is an implicit double sum over both µ and ν, and the derivatives
commute but F µν is antisymmetric. (Check this by interchanging the order
of summation.) Working out the components of this equation, we have
1 ∂J 0 ∂J 1 ∂J 2 ∂J 3
+
+
+
=0
c ∂t
∂x
∂y
∂z
(8.91)
which is just the charge conservation equation (8.80).
What about the remaining equations? Introduce the dual tensor Gµν
~ by B
~ and B
~ by − 1E,
~ resulting in
obtained from F µν by replacing 1c E
c
Guv
0

 −B x
=

y
 −B
−B z

Then the four equations
Bx
0
1
Ez
c
− 1c E y
By
− 1c E z
0
1
Ex
c
Bz

1
Ey 
c

− 1c E x 

0
∂Gµν
=0
∂xν

(8.92)
(8.93)
correspond to
−
~ ·B
~ = 0
∇
(8.94)
~
1 ∂B
1~
~ = 0
− ∇
×E
c ∂t
c
(8.95)
8.4. THE ELECTROMAGNETIC FIELD
55
which are precisely the two remaining Maxwell equations.
Some further properties of these tensors are
1
1 ~ 2
~ 2 = − 1 Gµν Gµν
+ |B|
Fµν F µν = − 2 |E|
2
c
2
1
1
~ ·B
~
Gµν F µν = − E
4
c
(8.96)
(8.97)
where care must be taken with the signs of the components of the covariant
tensors Fµν and Gµν . You may recognize these equations as corresponding
to important scalar invariants of the electromagnetic field.
Finally, it is possible to solve the sourcefree Maxwell equations by introducing a 4-potential
!
1
Φ
c
µ
A =
(8.98)
~
A
and defining
F µν =
∂Aµ
∂Aν
−
∂xµ
∂xν
(8.99)
where again care must be taken with the signs of the components with “downstairs” indices. Furthermore, the Lorentz force law can be rewritten in the
form
∂pµ
= quν F µν
(8.100)
m
∂τ
Note the appearance of the proper time τ in this equation. Just as in the previous chapter, this is because differentiation with respect to τ pulls through
a Lorentz transformation, which makes this a valid tensor equation, valid in
any inertial frame.